Topology, Geometry, and Group Theory

大约50年前，Aleksandrov， Busemann和Wald引入了定义比黎曼流形更一般度量空间“曲率”的上界和下界的思想。Aleksandrov主要对非负曲率及其在欧几里得三维凸多面体曲面上的应用感兴趣。大约十年前，当Gromov指出，由于拓扑和群论的原因，非正弯曲的情况比正弯曲的情况更有趣，主要原因是非正弯曲空间的全覆盖是可收缩的，人们对这个问题的兴趣重新燃起。Gromov还指出，这种空间有很多多面体的例子。特别是巴黎的建筑就是这样的例子。在过去的几年里，查尼教授和戴维斯教授合作研究非正弯曲多面体及其在群论中的应用。他们主要感兴趣的群体是考克斯特群体和马丁群体。这两种类型的组都与反射产生的组相关联。查尼和戴维斯也分别在这些领域进行了研究。他们将继续研究这些领域以及密切相关的主题，如Tits建筑及其自同构群的特性。由反射产生的群在数学和自然界的许多地方都有出现（例如，在晶体学中）。欧几里得平面的正则平铺的对称群就是这样的群。埃舍尔的绘画展示了非欧几里得平面几何中规则平铺的例子。这些也与反思组有关。查尼教授和戴维斯教授对这些更抽象的例子感兴趣。事实证明，抽象反射群（考克斯特群）总是可以被实现为非正弯曲空间的对称群，就像欧几里得平面和非欧几里得平面的情况一样。* * *

9803374 Davis About 50 years ago, Aleksandrov, Busemann and Wald introduced the idea of defining upper and lower bounds for the ``curvature'' of more general metric spaces than Riemannian manifolds. Aleksandrov was interested primarily in the case of nonnegative curvature and its applications to convex polyhedral surfaces in Euclidean 3-space. Interest in this subject was renewed about ten years ago when Gromov pointed out that for topological and group theoretic reasons the nonpositively curved case was much more interesting than the positively curved case, the main reason being that the universal cover of a nonpositively curved space is contractible. Gromov also pointed out that there were many polyhedral examples of such spaces. In particular, Tits buildings are such examples. For the past several years, Professors Charney and Davis have collaborated in investigating nonpositively curved polyhedra and applications to group theory. The groups in which they are primarily interested are Coxeter groups and Artin groups. Both types of groups are associated to groups generated by reflections. Both Charney and Davis have also accomplished separate research in these areas. They will continue their research on these areas as well as on closely related topics such as the properties of Tits buildings and their automorphism groups. Groups generated by reflections occur in many places in mathematics and in nature (for example, in crystallogrophy). The symmetry groups of regular tilings of the Euclidean plane are such groups. Escher's drawings show examples of regular tilings in non-Euclidean plane geometry. These are also associated with reflection groups. Professors Charney and Davis are interested in such examples in a more abstract setting. It turns out that abstract reflection groups (Coxeter groups) can always be realized as the symmetry groups of a space that is nonpositively curved, much as in the case of the Euclidean and non-Euclidean planes. ***